The \(x\)-ray transform for a non-Abelian connection in two dimensions (Q2747518)

After a useful introduction which provides physical as well as geometric motivations, the authors investigate a two dimensional integral geometry problem. Namely, they consider the following equation NEWLINE\[NEWLINE\theta\cdot \nabla C(x,\varphi) =A(x)\cdot\theta C(x,\varphi)\tag{*}NEWLINE\]NEWLINE where \(A(x)=(A_1(x), A_2(x))\) has compact support, where \(A_1,A_2\) are skew-Hermitian matrices of size \(m\), \(\theta= (\cos\varphi, \sin\varphi)\) and \(C(x,\varphi)= I_m(m\times m\) identity matrix) for \(x\cdot\theta \ll 0\). The problem is whether the knowledge of \(C(x,\theta)\) for \(x\cdot \theta\gg 0\) uniquely determine the coefficient matrices.NEWLINENEWLINENEWLINEThe question makes sense up to gauge transformations \(A\to G^{-1}A G-G^{-1}\nabla G\), where \(G(x)\) is a smooth function, taking values in the set of unitary matrices \(U(m)\), which is the identity outside the support of \(A\). This is because the matrix \(Y(x,\theta)\) such that \(C(x,\theta)= G(x)Y(x, \theta)\) satisfies the equation \(\theta\cdot \nabla Y(x,\varphi)= [G^{-1}AG-G^{-1}\nabla G]\cdot \theta Y(x,\varphi)\), and \(C=Y\) when the norm of \(x\) is sufficiently large.NEWLINENEWLINENEWLINEIn geometric terms, \(C\) is the parallel translation operator of the connection defined by -- \(A\) along straight lines, and the gauge transformation corresponds to the transformation law of the connection under frame changes. So the problem is the determination of \(A\) from the line integrals of the connection.NEWLINENEWLINENEWLINEThe authors obtain the following result:NEWLINENEWLINENEWLINETheorem 2.2. Suppose that \(C_1\) and \(C_2\) satisfy (*) with \(A\) replaced by \(A^{(1)}\) and \(A^{(2)}\), respectively, with the support of \(A^{(i)}\) in an open set \(\Omega\) with smooth boundary. Furthermore assume that \(C_1(x,\theta)= C_2 (x, \theta)=I\) for \(x\cdot\theta\ll 0\) and \(C_1(x,\theta)= C_2(x,\theta)\) for \(x\cdot \theta\gg 0\). If NEWLINE\[NEWLINE\frac 12\text{diam} (\Omega)^2(\sup\|K_{A^{(1)}} \|_{\text{op}}+ \sup\|K_{A^{(2)}} \|_{\text{op}}) <1,NEWLINE\]NEWLINE then \(A^{(1)}\) is gauge equivalent to \(A^{(2)}\). Here \(K_A\) is the curvature operator, and \(\|K_A\|_{\text{op}}\) the operator norm.